JEE MAIN - Physics (2025 - 28th January Evening Shift - No. 15)
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A bar magnet has total length $2 l=20$ units and the field point P is at a distance $\mathrm{d}=10$ units from the centre of the magnet. If the relative uncertainty of length measurement is $1 \%$, then uncertainty of the magnetic field at point P is :
Explanation
Without considering uncentainity in $\ell$.
$B=\frac{\mu_0}{4 \pi} \frac{m}{r^3}$
$\mathrm{B} \propto \frac{1}{\mathrm{r}^3}$
$$\frac{\Delta \mathrm{B}}{\mathrm{~B}}=3 \times\left(\frac{\Delta \mathrm{r}}{\mathrm{r}}\right)$$
$\%$ uncertainity in $\mathrm{B}=3 \%$
Method-2 :
With considering uncentainity in $\ell$.
$$\begin{aligned} & \mathrm{B} \propto \frac{1}{\mathrm{r}^3} \\ & \frac{\Delta \mathrm{~B}}{\mathrm{~B}}=\frac{\Delta \ell}{\ell}+3 \times\left(\frac{\Delta \mathrm{r}}{\mathrm{r}}\right)=1+3 \times 1=4 \% \end{aligned}$$
$\%$ uncertainity in $B=4 \%$
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